Differential Forms in Algebraic Topology

351 Pages·2011·14.23 MB·English

RaoulBo
Loring W. T
Iffere tia Forms
· Algebra·c
Topology
~
pringer
82
Graduate Texts in Mathematics
EditorialBoard
S.Axler F.W. Gehring P.R.Halmos
Springer
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Graduate Texts in Mathematics
TAKEunlZARlNG. Introductionto 33 HIRSCH. DifferentialTopology.
AxiomaticSetTheory.2nded. 34 SPITZER.PrinciplesofRandomWalk.
2 OXTOBV.MeasureandCategory.2nded. 2nded.
3 SCHAEFER.TopologicalVectorSpaces. 3S WERMER.BanachAlgebrasandSeveral
4 HILroN/STAMMBACH.ACoursein ComplexVariables.2nded.
Homological Algebra.2nded. 36 KELLEy/NAMIOKAetaleLinear
5 MACLANE.CategoriesfortheWorking TopologicalSpaces.
Mathematician. 37 MONK.MathematicalLogic.
6 HUGHES/PIPER. ProjectivePlanes. 38 GRAUERT/FRlTZSCHE.SeveralComplex
7 SERRE.ACourseinArithmetic. Variables.
8 TAKEtmlZARlNG.AxiomaticSetTheory. 39 ARVESON.AnInvitationtoC·-Algebras.
9 HUMPHREvs. IntroductiontoLieAlgebras 40 KEMENV/SNEuJKNAPP. Denumerable
andRepresentationTheory. MarkovChains.2nded.
10 COHEN.ACourseinSimpleHomotopy 41 APOSTOL. ModularFunctionsand
Theory. DirichletSeriesinNumberTheory.
II CONWAv. FunctionsofOneComplex 2nded.
VariableI.2nded. 42 SERRE.LinearRepresentationsofFinite
12 BEALS.AdvancedMathematicalAnalysis. Groups.
13 ANDERSON/FuLLER.RingsandCategories 43 GILLMAN/JERJSON. RingsofContinuous
ofModules.2nded. Functions.
14 GOLUBITSKV/GUILLEMIN.StableMappings 44 KENDIG.ElementaryAlgebraicGeometry.
andTheirSingularities. 45 LOEVE.ProbabilityTheoryI.4thed.
15 BERBERlAN.LecturesinFunctional 46 LOEVE.ProbabilityTheory II. 4thed.
AnalysisandOperatorTheory. 47 MOISE.GeometricTopologyin
16 WINTER.TheStructureofFields. Dimensions2and3.
17 ROSENBLAlT.RandomProcesses.2nded. 48 SACHslWu.General Relativityfor
18 HALMos. MeasureTheory. Mathematicians.
19 HALMos.AHilbertSpaceProblemBook. 49 GRUENBERG/WEIR.LinearGeometry.
2nded. 2nded.
20 HUSEMOLLER. FibreBundles.3rded. 50 EDWARDS.Fennat'sLastTheorem.
21 HUMPHREYS.LinearAlgebraicGroups. 51 KUNGENBERG.ACourseinDifferential
22 BARNES/MACK.AnAlgebraicIntroduc~ion Geometry.
toMathematical Logic. 52 HARTSHORNE.AlgebraicGeometry.
23 GREUB.LinearAlgebra.4thed. 53 MANIN.ACourseinMathematicalLogic.
24 HOLMES.GeometricFunctionalAnalysis 54 GRAVERlWATKINS.Combinatoricswith
andItsApplications. EmphasisontheTheoryofGraphs.
25 HEwm/STRoMBERG.Real andAbstract 55 BRowN/PEARCY.IntroductiontoOperator
Analysis. Theory I: ElementsofFunctional
26 MANES. AlgebraicTheories. Analysis.
27 KELLEY.GeneralTopology. 56 MASSEY.AlgebraicTopology: An
28 ZARISKIISAMUEL.CommutativeAlgebra. Introduction.
Vol.1. 57 CROWELuFox.IntroductiontoKnot
29 ZARlSKIISAMUEL.CommutativeAlgebra. Theory.
Vol.lI. 58 KOBun.p-adicNumbers,p-adic
30 JACOBSON. LecturesinAbstractAlgebraI. Analysis,andZeta-Functions.~nded.
BasicCOnCepl4t. S9 LANG.CyclotomicFields.
31 JACOBSON.LecturesinAbstractAlgebra 60 ARNOLD. MathematicalMethodsin
II. LinearAlgebra. Classical Mechanics.2nded.
32 JACOBSON. LecturesinAbstractAlgebra
III.TheoryofFieldsandGaloisTheory. COnlinuedafterindex
Raoul Bott Loring W. Tu
Differential Forms in
Algebraic Topology
With92Illustrations
Springer
ro
~ f· ~.. t }J'-..~.. .iJ
RaoulBott LoringW.Tu
MathematicsDepartment DepartmentofMathematics
HarvardUniversity TuftsUniversity
Cambridge,MA02138-2901 Medford, MA02155-7049
USA USA
EditorialBoard
S.Axler F.W.Gehring P.R.Halmos
Departmentof Departmentof Departmentof
Mathematics Mathematics Mathematics
MichiganStateUniversity UniversityofMichigan SantaClaraUniversity
EastLansing,MI48824 AnnArbor, MI48109 SantaClara,CA
95053
USA USA USA
MathematicsSubjectClassifications(1991):S7Rxx,58Axx,14F40
LibraryofCongressCataloging-in-PublicationData
Bott,Raoul,1924-
Differentialformsinalgebraictopology
(Graduatetextsinmathematics:82)
Bibliography:p.
Includesindex.
1. Differentialtopology. 2. Algebraic
topology. 3. Differentialforms: I. Tu
w.
Loring ~I. Title. III. Series.
QA613.6.B67 514'.72 81-9172
Printedonacid-freepaper.
©1982Springer-VerlagNewYork,Inc.
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ReprintedinChinabyBeijingWorldPublishingCorporation,1999
98765 4
ISBN0-387-90613-4 Springer-Verlag NewYork Berlin Heidelberg
ISBN3-540-90613-4 Springer-Verlag Berlin Heidelberg New¥ork SPIN 10635035
For
Phyllis Bott
and
Lichu and Tsuchih Tu

Preface
The guiding principle in this book is to use differential forms as an aid in
exploring some ofthe less digestible aspects ofalgebraic topology. Accord
ingly,.we move primarily in the realm of smooth manifolds and use the
de Rham theory as a prototype ofall ofcohomology. For applications to
homotopy theory we also discuss by way of analogy cohomology with
arbitrarycoefficients.
Although we have in mind an audience with prior exposure to algebraic
or differential topology, for the most part a good knowledge of linear
algebra, advanced calculus, and point·set topology should suffice. Some
acquaintance with manifolds, simplicial complexes, singular homology and
cohomology, and homotopy groups is helpful, but not really necessary.
Within the text itself we have stated with care the more advanced results
that are needed, so that a mathematically mature reader who aCcepts these
background materials on faith should be able to read the entire book with
theminimalprerequisites. .
There are more materials here than can be reasonably covered in a
one-semestercourse. Certain sections may be omitted at first reading with
out loss of continuity. We have indicated these in the schematic diagram
thatfollows. .
This book is.not intended to befoundational; rather, it is only meant to
open some of the doors to the formidable edifice of modem algebraic
topology. We offer it in the hope that such an informal account of the
subjectatasemi-introductorylevelfills a gapintheliterature.
It would be impOssible to mention all the friends, colleagues, and
students whose ideas have contributed to this book. But the senior
author would like on this occasion to express his deep gratitude, rust
of all to his primary topology teachers E. Specker, N. Steenrod, and
vii
viii Preface
K.Reidemeisterofthirtyyearsago,andsecondlytoH.Samelson,A.Shapiro,
I. Singer, J.-P. Serre, F. Hirzebruch, A. Borel, J. Milnor, M. Atiyah, S.-s.
Chern, J. Mather, P. Baum, D. Sullivan, A. Haefliger, and Graeme Segal,
who, mostly incollaboration, have continued this word ofmouth education
to the present; the junior author is indebted to Allen Hatcher for having
initiated him into algebraic topology. The reader will find their influence if
not in all, then certainly in the more laudable aspects ofthis book. We also
owe-thanks to the many other people who have helped with our project: to
Ron Donagi, Zbig Fiedorowicz, Dan Freed, Nancy Hingston, and Deane
Yang for their reading of various portions ofthe manuscript and for their
critical comments, to Ruby Aguirre, Lu Ann Custer, Barbara Moody, and
Caroline Underwoodfor typing services, and to thestaffofSpringer-Verlag
for itspatience,dedication,andskill.
For the Revised Third Printing
Whilekeepingthetextessentially thesame as inprevious printings,we have
made numerous local changes throughout. The more significant revisions
concernthecomputationoftheEulerclassinExample6.44.1 (pp.75-76),the
proofofProposition 7.5 (p. 85), the treatment ofconstant and locally con
stantpresheaves(p. 109and p. 143),the proofofProposition 11.2(p. 115),a
localfinite hypothesis onthe generalized Mayer-Vietorissequencefor com
pact supports (p. 139), transgressive elements (Prop. 18.13, p. 248), and the
discussion ofclassifyingspacesfor vectorbundles(pp. 297-3(0).
We would like to thank Robert Lyons, Jonathan Dorfman, Peter Law,
Peter Landweber, and Michael Maltenfort, whose lists ofcorrections have
beenincorporatedintothesecond and thirdprintings.
RAOULBOTT
LORINGTu
Interdependence of the Sections
1-6
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8-11
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13-16 20-22
17 23
18
19
ix